A ROBUST ESTIMATION METHOD OF LOCATION AND SCALE WITH APPLICATION IN MONITORING PROCESS VARIABILITY ROHAYU BT MOHD SALLEH

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1 A ROBUST ESTIMATION METHOD OF LOCATION AND SCALE WITH APPLICATION IN MONITORING PROCESS VARIABILITY ROHAYU BT MOHD SALLEH A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy (Mathematics) Faculty of Science Universiti Teknologi Malaysia AUGUST 2013

2 iii This thesis is dedicated to Pak Salleh and Mak Yam for their loves, endless support and encouragement.

3 iv ACKNOWLEDGEMENT All Praise and Gratitude is due to Allah SWT, the Most Gracious, the Most Merciful. For without Him, everything will be cease to be, and peace and blessings upon our beloved prophet Muhammad SAW. Though only my name appears on the cover of this thesis, a great many people have contributed to its realization. It is inspiring to meet someone whom wakes you up from your dream sometimes to remind you to keep focus and never stop getting inspired and to go back on track. Sometimes life, dilemmas, responsibilities, lack of time and sitting in my comfort zone makes me forget what my real intention is. I m thankful and grateful that god sends me these people who are like angels to me those truly inspire me in order to inspire others. My main supervisor, Prof. Dr. Maman A. Djauhari, for his wisdom, knowledge and commitment to the highest standard, inspired and motivated me. He read each chapter of this thesis few times, which finally transformed it from a hardly understandable draft to a readable thesis. I greatly appreciate his patience in reviewing and providing constructive criticism. My co-supervisors, Assoc. Prof. Dr. Robiah Adnan and Assoc. Prof Dr Dyah Erny Herwindiati, for their helpful suggestions, moral support and constant encouragement. My friends that helped me to stay positive thorough these years. I also would like to sincerely thank and acknowledge the significant role that helping this thesis to materialize, Minister of Higher Education and Universiti Tun Hussein Onn Malaysia, the scholarship given is gratefully appreciated and also the administrative staffs at Department of Mathematical Sciences, UTM. Also, very special thanks to each person in my families especially to my parents and siblings for their great patience and moral support during my PhD study. Thank you.

4 v ABSTRACT This thesis consists of two parts; theoretical and application. The first part proposes the development of a new method for robust estimation of location and scale, in data concentration step (C-step), of the most widely used method known as fast minimum covariance determinant (FMCD). This new method is as effective as FMCD and minimum vector variance (MVV) but with lower computational complexity. In FMCD, the optimality criterion of C-step is still quite cumbersome if the number of variables p is large because of the computation of sample generalized variance. This is the reason why MVV has been introduced. The computational 3 2 complexity of the C-step in FMCD is of order O( p ) while MVV is O( p ). This is a significant improvement especially for the case when p is large. In this case, although MVV is faster than FMCD, it is still time consuming. Thus, this is the principal motivation of this thesis, that is, to find another optimal criterion which is of far higher computational efficiency. In this study, two other different optimal criteria which will be able to reduce the running time of C-step is proposed. These criteria are (i) the covariance matrix equality and (ii) index set equality. Both criteria do not require any statistical computations, including the generalized variance in FMCD and vector variance in MVV. Since only a logical test is needed, the computational complexities of the C-step are of order O( p ln p ). The second part is the application of the proposed criteria in robust Phase I operation of multivariate process variability based on individual observations. Besides that, to construct a more sensitive Phase II operation, both Wilks W statistic and Djauhari s F statistic are used. Both statistics have different distributions and is used to measure the effect of an additional observation on covariance structure.

5 vi ABSTRAK Tesis ini mengandungi dua bahagian; teori dan aplikasi. Bahagian pertama mencadangkan pembangunan kaedah baru untuk penganggaran teguh lokasi dan skala, dalam langkah penumpuan data (C-langkah), dari kaedah yang paling digunakan secara meluas dikenali sebagai penentu kovarians minimum cepat (FMCD). Kaedah baru ini efektif seperti FMCD dan varians vektor minimum (MVV) tetapi kerumitan pengiraannya adalah rendah. Dalam FMCD, secara optimum kriteria bagi C-langkah masih agak rumit jika bilangan pembolehubah p adalah besar disebabkan pengiraan sampel varians teritlak. Inilah alasan mengapa MVV diperkenalkan. Kerumitan pengiraan C-langkah dalam FMCD adalah 3 2 peringkat O( p ) manakala MVV adalah O( p ). Ini adalah satu peningkatan yang bererti terutamanya untuk kes bila p besar. Dalam kes ini, walaupun MVV lebih cepat daripada FMCD, pengiraannya masih mengambil masa. Oleh itu, motivasi utama tesis ini ialah untuk mencari kriteria optimum yang lain dimana pengiraannya jauh lebih efisien. Dalam kajian ini, dua kriteria optimum yang berbeza yang boleh mengurangkan masa pengiraan di dalam C-langkah dicadangkan. Kriteria tersebut adalah (i) kesaksamaan kovarians matrik dan (ii) kesaksamaan set indeks. Kedua-dua kriteria ini tidak memerlukan sebarang pengiraan statistik, termasuklah varians teritlak dalam FMCD dan varians vektor dalam MVV. Disebabkan hanya ujian logik diperlukan, kerumitan pengiraan bagi C-langkah adalah peringkat O( p ln p ). Bahagian kedua adalah pengunaan kriteria yang dicadangkan dalam Fasa I dalam pemantauan kepelbagaian proses multivariat secara teguh berdasarkan sampel individu. Selain itu, untuk membina operasi Fasa II yang lebih sensitif, kedua-dua statistik W daripada Wilks dan statistik F daripada Djauhari digunakan. Kedua dua statistik mempunyai taburan yang berbeza dan digunakan untuk mengukur kesan penambahan data pada struktur kovarians.